Optimal. Leaf size=52 \[ -\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.0834772, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 12.9587, size = 44, normalized size = 0.85 \[ \frac{c d}{e^{3} \left (d + e x\right )^{2}} - \frac{c}{e^{3} \left (d + e x\right )} - \frac{a e^{2} + c d^{2}}{3 e^{3} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.026372, size = 39, normalized size = 0.75 \[ -\frac{a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(d + e*x)^4,x]
[Out]
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Maple [A] time = 0.007, size = 51, normalized size = 1. \[{\frac{cd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}+c{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.703888, size = 85, normalized size = 1.63 \[ -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200986, size = 85, normalized size = 1.63 \[ -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.34213, size = 66, normalized size = 1.27 \[ - \frac{a e^{2} + c d^{2} + 3 c d e x + 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.212136, size = 50, normalized size = 0.96 \[ -\frac{{\left (3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^4,x, algorithm="giac")
[Out]